The moduli space of Higgs bundles on a compact Riemann surface C for a group G is diffeomorphic to the character variety of representations 

of the fundamental group in G. One description depends on the complex structure of C, the other is purely topological. Using a natural symplectic Ehresmann connection we show how to build the complex structure on the family of Higgs bundle moduli spaces over Teichmuller space and derive some consequences for the energy of the associated harmonic maps.